scholarly journals Inferring the instability of a dynamical system from the skill of data assimilation exercises

2021 ◽  
Vol 28 (4) ◽  
pp. 633-649
Author(s):  
Yumeng Chen ◽  
Alberto Carrassi ◽  
Valerio Lucarini
Keyword(s):  
Dynamical System ◽  
Data Assimilation ◽  
Observational Data ◽  
Initial Conditions ◽  
Model Error ◽  
High Dimensional ◽  
Accurate Knowledge ◽  
Sensitive Dependence ◽  
Abstract Data ◽  
Very High

Abstract. Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov–Sinai entropy and perform numerical experiments on the Vissio–Lucarini 2020 model, a recently proposed generalization of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

scholarly journals Inferring the instability of a dynamical system from the skill of data assimilation exercises

2021 ◽  
Author(s):  
Yumeng Chen ◽  
Alberto Carrassi ◽  
Valerio Lucarini
Keyword(s):  
Dynamical System ◽  
Data Assimilation ◽  
Observational Data ◽  
Initial Conditions ◽  
Model Error ◽  
High Dimensional ◽  
Accurate Knowledge ◽  
Sensitive Dependence ◽  
Abstract Data ◽  
Very High

Abstract. Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error, and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov-Sinai entropy, and perform numerical experiments on the Vissio-Lucarini 2020 model, a recently proposed generalisation of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

TREATMENT OF THE ERROR DUE TO UNRESOLVED SCALES IN SEQUENTIAL DATA ASSIMILATION

2011 ◽  
Vol 21 (12) ◽  
pp. 3619-3626 ◽  
Author(s):  
ALBERTO CARRASSI ◽  
STÉPHANE VANNITSEM
Keyword(s):  
Dynamical System ◽  
Kalman Filter ◽  
Data Assimilation ◽  
Large Scale ◽  
Model Error ◽  
Environmental Modeling ◽  
Sequential Data ◽  
Chaotic Dynamical System ◽  
Error Covariance ◽  
Sequential Data Assimilation

In this paper, a method to account for model error due to unresolved scales in sequential data assimilation, is proposed. An equation for the model error covariance required in the extended Kalman filter update is derived along with an approximation suitable for application with large scale dynamics typical in environmental modeling. This approach is tested in the context of a low order chaotic dynamical system. The results show that the filter skill is significantly improved by implementing the proposed scheme for the treatment of the unresolved scales.

scholarly journals ‘Intractable and unsolved’: some thoughts on statistical data assimilation with uncertain static parameters

2013 ◽  
Vol 371 (1991) ◽  
pp. 20120297 ◽  
Author(s):  
Jonathan Rougier
Keyword(s):  
Numerical Methods ◽  
Data Assimilation ◽  
State Space ◽  
Stochastic Models ◽  
Environmental Science ◽  
Statistical Data ◽  
Initial Conditions ◽  
The State ◽  
Likelihood Functions ◽  
Sensitive Dependence

If you seem to be able to do data assimilation with uncertain static parameters then you are probably not working in environmental science. In this field, applications are often characterized by sensitive dependence on initial conditions and attracting sets in the state-space, which, taken together, can be a major challenge to numerical methods, leading to very peaky likelihood functions. Inherently stochastic models and uncertain static parameters increase the challenge.

Coupled data assimilation in climate research: A brief review of applications in ocean and land

2018 ◽  
Vol 3 (2) ◽  
Author(s):  
Kazuyoshi Suzuki 1 ◽  
Milija ZUPANSKI 2
Keyword(s):  
Data Assimilation ◽  
Observational Data ◽  
Initial Conditions ◽  
Monitoring Networks ◽  
Coupled Models ◽  
Climate Change Research ◽  
Empirical Parameter ◽  
Strongly Coupled ◽  
Weakly Coupled ◽  
Coupled Data Assimilation

Regions of the cryosphere, including the poles, that are currently unmonitored are expanding, therefore increasing the importance of satellite observations for such regions. With the increasing availability of satellite data in recent years, data assimilation research that combines forecasting models with observational data has begun to flourish. Coupled land/ice-atmosphere/ocean models generally improve the forecasting ability of models. Data assimilation plays an important role in such coupled models, by providing initial conditions and/or empirical parameter estimation. Coupled data assimilation can generally be divided into three types: uncoupled, weakly coupled, or strongly coupled. This review provides an overview of coupled data assimilation, introduces examples of its use in research on sea ice-ocean interactions and the land, and discusses its future outlook. Assimilation of coupled data constitutes an effective method for monitoring cold regions for which observational data are scarce and should prove useful for climate change research and the design of efficient monitoring networks in the future.

On the Iteration Properties of Large Deviations Theorem

2016 ◽  
Vol 26 (03) ◽  
pp. 1650054 ◽  
Author(s):  
Xinxing Wu ◽  
Xiong Wang
Keyword(s):  
Dynamical System ◽  
Positive Integer ◽  
Large Deviations ◽  
Initial Conditions ◽  
Ergodic Properties ◽  
Sensitive Dependence

An important feature of chaoticity of a dynamical system [Formula: see text] is its sensitive dependence on initial conditions, which has recently been extended to a concept of ergodic sensitivity. This paper proves that if there exists a positive integer [Formula: see text] such that [Formula: see text] satisfies the large deviations theorem or ergodicity, then so is [Formula: see text]. Moreover, the iteration invariance of some ergodic properties are obtained.

scholarly journals The role of optimal perturbations in improvement of stabil- ity and forecasting capacity of adaptive filter for high di- mensional systems

2018 ◽  
Vol 210 ◽  
pp. 02033
Author(s):  
Hong Son Hoang ◽  
Rémy Baraille
Keyword(s):  
Dynamical System ◽  
Invariant Subspace ◽  
Adaptive Filter ◽  
Numerical Algorithms ◽  
High Dimensional ◽  
Optimal Perturbation ◽  
Different Types ◽  
Theoretical Results ◽  
Very High

This paper is devoted to an optimal perturbation (OP) which allows to qualify a capacity of the dynamical system to predict more or less correctly the system behaviour in the future. The different types of OPs, deterministic, stochastic, OP in an invariant subspace, are introduced. The theoretical results on the optimality of the introduced OPs are presented. The different simple and efficient numerical algorithms for computation of the OPs are outlined which constitute a basis for implementation of a stable adaptive filter in a very high dimensional environment.

scholarly journals Aggregated Negative Feedback in a Generalized Lorenz Model

2019 ◽  
Vol 29 (03) ◽  
pp. 1950037 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Steady State ◽  
Negative Feedback ◽  
Critical Points ◽  
Initial Conditions ◽  
High Dimensional ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Higher Dimensional ◽  
Sensitive Dependence ◽  
Steady State Solutions

In this study, we first present a generalized Lorenz model (LM) with [Formula: see text] modes, where [Formula: see text] is an odd number that is greater than three. The generalized LM (GLM) is derived based on a successive extension of the nonlinear feedback loop (NFL) with additional high wavenumber modes. By performing a linear stability analysis with [Formula: see text] and [Formula: see text], we illustrate that: (1) within the 3D, 5D, and 7D LMs, the appearance of unstable nontrivial critical points requires a larger Rayleigh parameter in a higher-dimensional LM and (2) within the 9DLM, nontrivial critical points are stable. By comparing the GLM with various numbers of modes, we discuss the aggregated negative feedback enabled by the extended NFL and its role in stabilizing solutions in high-dimensional LMs. Our analysis indicates that the 9DLM is the lowest order generalized LM with stable nontrivial critical points for all Rayleigh parameters greater than one. As shown by calculations of the ensemble Lyapunov exponent, the 9DLM still produces chaotic solutions. Within the 9DLM, a larger critical value for the Rayleigh parameter, [Formula: see text], is required for the onset of chaos as compared to a [Formula: see text] for the 3DLM, a [Formula: see text] for the 5DLM, and a [Formula: see text] for the 7DLM. In association with stable nontrivial critical points that may lead to steady-state solutions, the appearance of chaotic orbits indicates the important role of a saddle point at the origin in producing the sensitive dependence of solutions on initial conditions. The 9DLM displays the coexistence of chaotic and steady-state solutions at moderate Rayleigh parameters and the coexistence of limit cycle and steady-state solutions at large Rayleigh parameters. The first kind of coexistence appears within a smaller range of Rayleigh parameters in lower-dimensional LMs (i.e. [Formula: see text] within the 3DLM) but in a wider range of Rayleigh parameters within the 9DLM (i.e. [Formula: see text]). The second kind of coexistence has never been reported in high-dimensional Lorenz systems.

Atmospheric Data Assimilation with an Ensemble Kalman Filter: Results with Real Observations

2005 ◽  
Vol 133 (3) ◽  
pp. 604-620 ◽  
Author(s):  
P. L. Houtekamer ◽  
Herschel L. Mitchell ◽  
Gérard Pellerin ◽  
Mark Buehner ◽  
Martin Charron ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Forecast Error ◽  
Lower Troposphere ◽  
Model Error ◽  
Atmospheric Data ◽  
Ensemble Mean ◽  
Parameterized Model

Abstract An ensemble Kalman filter (EnKF) has been implemented for atmospheric data assimilation. It assimilates observations from a fairly complete observational network with a forecast model that includes a standard operational set of physical parameterizations. To obtain reasonable results with a limited number of ensemble members, severe horizontal and vertical covariance localizations have been used. It is observed that the error growth in the data assimilation cycle is mainly due to model error. An isotropic parameterization, similar to the forecast-error parameterization in variational algorithms, is used to represent model error. After some adjustment, it is possible to obtain innovation statistics that agree with the ensemble-based estimate of the innovation amplitudes for winds and temperature. Currently, no model error is added for the humidity variable, and, consequently, the ensemble spread for humidity is too small. After about 5 days of cycling, fairly stable global filter statistics are obtained with no sign of filter divergence. The quality of the ensemble mean background field, as verified using radiosonde observations, is similar to that obtained using a 3D variational procedure. In part, this is likely due to the form chosen for the parameterized model error. Nevertheless, the degree of similarity is surprising given that the background-error statistics used by the two procedures are rather different, with generally larger background errors being used by the variational scheme. A set of 5-day integrations has been started from the ensemble of initial conditions provided by the EnKF. For the middle and lower troposphere, the growth rates of the perturbations are somewhat smaller than the growth rate of the actual ensemble mean error. For the upper levels, the perturbation patterns decay for about 3 days as a consequence of diffusive model dynamics. These decaying perturbations tend to severely underestimate the actual error that grows rapidly near the model top.

scholarly journals The Effect of the Equivalent-Weights Particle Filter on Dynamical Balance in a Primitive Equation Model

2015 ◽  
Vol 143 (2) ◽  
pp. 581-596 ◽  
Author(s):  
Melanie Ades ◽  
Peter Jan van Leeuwen
Keyword(s):  
Probability Density Function ◽  
Data Assimilation ◽  
Particle Filter ◽  
Gravity Wave ◽  
Wave Energy ◽  
Initial Conditions ◽  
Model Error ◽  
Equation Model ◽  
Primitive Equation ◽  
Dynamical Balance

Abstract The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed. The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed. This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

scholarly journals A Study of Chaos in Dynamical Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
S. Effah-Poku ◽  
W. Obeng-Denteh ◽  
I. K. Dontwi
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Initial Conditions ◽  
Period Doubling ◽  
Dense Orbit ◽  
Routes To Chaos ◽  
Sensitive Dependence ◽  
Nonlinear Science

The behavior of systems such as periodicity, fixed points, and most importantly chaos has evolved as an integral part of mathematics, especially in dynamical system. This research presents a study on chaos as a property of nonlinear science. Systems with at least two of the following properties are considered to be chaotic in a certain sense: bifurcation and period doubling, period three, transitivity and dense orbit, sensitive dependence to initial conditions, and expansivity. These are termed as the routes to chaos.

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